In this section, we recall some basic facts and estimates of asymptotically Euclidean 3-manifold. Let \((M^3, g)\)  be a complete AE 3-manifold. By Lemma 4.1 in [1]}, we know inside \(M\)  there is a trapped compact region \(T\)  whose topological boundary consists of smooth embedded minimal 2-spheres. The exterior region \(M_{ext}\)  is defined as the metric completion of a connected component of \(M \setminus T\)  associated to one end. Then \(M_{ext}\)  is connected and AE, has a compact minimal boundary, and contains no other compact minimal surfaces (even immersed).
